October 2005

ON REAL ECONOMIC FREEDOM

by Serge-Christophe KOLM*

Abstract

A priori, real economic freedom is purchasing (and selling) power. Yet, Xu’s theorem

comforts ranking the freedom of choice provided by budget sets as their volume in deriving it

from three axioms. However, one and a half of these axioms can be discussed. In contrast, the

standard measure of purchasing power leads one to order the freedom provided by budget sets

as the distance to the origin of the intersection of the budget hyperplanes with a given ray

from the origin. Hence, equal budget freedoms correspond to pencils of budget hyperplanes.

Applied to labour and earnings of individuals with different wage rates, this equal freedom

yields the distributive principle of equal labour income equalization.

Keywords: freedom, freedom of choice, economic freedom, purchasing power, budget

ranking, equality.

JEL classification numbers: D31, D46, D63, H21, J33

1. Economic freedom

Economic freedom often means freedom of exchange. Freedom of exchange is a part of basic

freedoms or basic rights, which constitute the legal and moral basis of our societies. It has

often been reproached to these rights that they may leave you with little actual freedom if you

do not have the means to make use of them. In particular, even if all “men are free and equal

in rights” (the 1789 Declaration), this allows for very unequal actual freedom. “Rich and poor

are equally free to sleep under bridges” (Anatole France). Marx denounces these basic rights

* CREM, IDEP, EHESS.

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as being only “formal freedom” and not providing “real freedom”.1 Yet, he thus inspired

suppressions of “formal freedoms” with the historical dramatic consequences we know. Real

freedom requires both formal freedom and other means, that is, with free exchange, income or

means to acquire it such as a sufficient wage rate. The budget sets are the domains of possible

choice which provide the corresponding “freedom of choice.”

One can certainly admit that, with given prices, a higher income provides a higher

such freedom. At least, it provides a possibility set that includes the other in adding

possibilities to have more of all goods. As common language puts it: “purchasing power” is

higher. Yet, what can be said when prices are not the same and the budget sets are not related

by inclusion? This situation is a very common problem for economists and economic

statisticians. They face it in choosing a price index for computing a “purchasing power” in

dividing income by this index. The result is also “real” in another sense of the term, now

opposed to “nominal”, i.e., it does not change if incomes and prices are multiplied by the

same number (for whatever reason). More precisely, the price index is a linear function of

prices, with coefficients that represent quantities of goods in a “basket” the choice of which is

the subject of the theory of price indexes. The result thus measures the purchasing power of

an income facing a set of prices, hence of the corresponding budget set, as the number of such

baskets that this income can buy when facing these prices. That is to say, this classical real

freedom is the distance to the origin of the intersect of the budget hyperplane with the ray

from the origin bearing the vector of the coefficients of the price index. One consequence is

that the equality of such freedoms means that the budget hyperplanes pass through the same

point which represents the vector of the price-index basket multiplied by the same number

(they constitute a “pencil” of hyperplanes).

This comparison of purchasing powers is often done for the different prices at

different dates or in different places. Yet, there is a still more important application, since

freedom is a priori that of individuals, and different individuals generally face different prices

in free exchange for the good which can be said to be the most important one, labour, whose

price, the wage rate, depends on the different productive capacities and on the demands for

them, and is also the market price of the good leisure. This leads to the determination of given

incomes, or income transfers, that maintain an equal purchasing power, or real economic 1 Marx also objected to the individualism of basic rights, thus echoing a reproach made by Robespierre as early as 1789.

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freedom, for individuals endowed with different given productive capacities, hence facing

different wage rates. The result in the simplest case turns out to be that each individual i

receives the net transfer ti=k·( w –wi), where wi is individual i’s wage rate, w =(1/m)Σwi is the

average wage rate for m individuals, k is a number defining the price index, and ti is a subsidy

if ti>0 and a tax of –ti if ti<0 (see Section 4 below).2

However, a remarkable recent article of Yongsheng Xu (2004) proposes a strong

axiomatic basis for the application to competitive budget sets of an old alternative proposal,

which consists of ranking the freedom offered by domains of choice of quantities of goods by

their volume. This is at odds with the foregoing result based on standard practice. For

instance, in the case of two goods, the budget lines corresponding to equal real freedom pass

through the same point with the standard comparison and are tangent to the same rectangular

hyperbola with the volume ranking.

In 1972, Jean-Marc Oury, then a student, proposed to me, as topic of dissertation, the

measure of the freedom of choice of a bundle of commodities by the volume of the possibility

set. I rather discouraged him. One of my reasons was that this amounts to measuring real

income with a price index which is the geometric mean of prices, and this neat property has

strange consequences shortly recalled – and needs to be justified.3 However, Xu derives his

conclusion from axioms, not from consequences.4 The volume ranking was nevertheless

among the rankings or measures of freedom of choice whose properties I analyzed.5 Yet, the

contradiction has to be elucidated at the level of the axioms and of their meaning.6

Most generally, there are three kinds of freedoms, which can be labelled negative

freedom, positive freedom, and mental freedom. They are often closely interrelated. Negative 2 And Kolm 2004a. 3 Moreover, forbidding access to any one good makes you less free than any other kind of constraint. Yet, this is not a budget constraint, and hence this could be objected to Oury but cannot to Xu. 4 Classical epistemology states that rules should be judged jointly according to their statements, their conditions (including axioms), and their consequences (see, e.g., Plato’s “dialectic” in Republic or Rawls’s “reflective equilibrium”). 5 See Kolm 2004a, Chap. 24 (also 1993). 6 Discouraged, Jean-Marc Oury abandoned his Ph.D. Hired by the French industry mogul Guy Dejouany at the Compagnie Générale des Eaux, he became in charge of the real-estate branch where he drove the Maisons Phenix to buoyancy and then to failure. This did not discourage Dejouany from hiring other bright young people from the best school. The next one was Jean-Marie Messier, who had Oury fired, took over the firm, and did exactly the same thing in the media business while renaming the firm Vivendi Universal.

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freedom or social freedom is freedom from the forceful interference of other humans. It is

defined by the nature of the constraints and it consists of the classical basic rights or liberties.

In economics, it consists of free enterprise and free exchange and market. People are of course

constrained not to forcefully interfere with others, in particular to respect the consequences of

free actions or agreements (such as rights so created). Social freedom raises no issue of rivalry

and can be at satiety (remaining rivalry concerns the allocation of rights concerning given

resources). Positive freedom is defined by the domains of free choice, and it also is the real

freedom of the discussions referred to above. In a market system, it is aptly called purchasing

power (and selling power). Mental freedom is the freedom to determine one’s emotions and

preferences. The “autonomy” of Rousseau and Kant is a type of it. In economics it would

refer to issues such as an absence of manipulation by advertisement, deliberative choices, or

choice of criteria of fairness.7 The present topic fully discards preferences and hence mental

freedom. It is concerned with the second kind of liberty, positive freedom, in the context of

the economic manifestation of the first, free markets. That is, it concerns the purchasing

power of budget sets, or budget freedom.8

Section 2 considers Xu’s theorem. The consequence of the alternative of classical

price indexes is shown in Section 3. Section 4 applies this result to the question of

macrojustice.9

2. Freedom as volume

The freedom of choice offered by domains of choice (possibility or opportunity sets) D will

be ordered by an ordering assumed here to be representable by an ordinal freedom function

7 One can also build an economic model of consciously influencing one’s preferences (see Kolm 1982, chap.23, 1985). 8 The vocabulary of freedom has both this established base and fluctuations. For instance, what Isaiah Berlin (1958) calls positive freedom is largely mental freedom violated by ideologies. General mental freedom, including the basic issue of mastering one’s desires which in principle could have drastic economic consequences, is the topic of my book Happiness-Freedom (Deep Buddhism and Modernity) (1982, in French). 9 I wish to thank Nicolas Gravel for very usefull corrections and suggestions.

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F(D) (this representability is no restriction for the present problem).10 One property is

).()( DFDFDD ≤′⇒⊆′ 11

We consider particular D’s which are budget sets. Denote as i one of n goods, xi≥0 its

quantity, x={xi}∈ n+ℜ a bundle of these goods, pi>0 a constant price of good i, p={pi} the

price vector, and y an income. In the space of x, Σpixi=y is the equation of the budget

hyperplane, and Σpixi≤ y with xi≥0 for all i defines the budget set. Then, ai=y/pi is the xi of the

intersect of the budget hyperplane with the axis of the xi, and a={ai} denotes the set or vector

of the ai’s.

The budget set is defined by the pair (y, p), and the freedom function can be written as

F(y, p). A priori, this function represents a real issue (not a nominal one), and hence it is

homogeneous of degree zero in y and p. Hence,

F(y, p)=F(1, 1/a1, ..., 1/an)=φ(a).

Function F will be taken as increasing in y (and decreasing in the pi) since an increase

in y (and a decrease in a pi) add bundles containing larger quantities of all goods than previous

possible ones. Hence function φ is increasing in the ai.

The following remarks in this section are standard or obvious.

If functions F and φ are such that their ordering of the a does not change if the same

coordinate of all a is multiplied by the same positive number, for all coordinates and numbers,

that is,

φ(a)≥φ( a′ )⇒φ(a1,...,λai,..., an)≥φ( 1a′ ,...,λ 1a′ ,..., na′ ) (1)

for all a, i, and λ>0, then φ is of the form

φ(a)=φ1( Π iiaα ) (2)

10 Xu’s ordering is so representable. He has studied various possible general properties of freedom orderings and functions in Pattanaik and Xu (2000). 11 This property seems unavoidable if the various costs of choice are considered a different issue from freedom (including material costs, mental costs, disliking responsibility, and anguish of choice – emphasized by Kierkegaard and Sartre).

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where Π denotes the product, αi>0 are numbers, and φ1 is an increasing function; and

conversely.12

Then, function φ(a) with this form is symmetrical if and only if all the αi are equal:

αi=α>0 for all i. Then,

φ(a)=φ2( Π ia )

where φ2 is an increasing function.

Yet, the volume of the budget set is V=(1/n!) Π ia . Hence,

φ(a)=φ3 (V)

where φ3 is an increasing function.

Xu postulates the symmetry of the function φ(a). This amounts to the symmetry of the

freedom function F(y, p) in the prices. Then, your freedom of choice is not changed by a

permutation of the prices of the goods. It is not changed if you pay a sandwich for the price of

a car and conversely. The mad store manager who assigns randomly his price tags to the

goods leaves his customers as free. Moreover, if the price of one good doubles, your freedom

is the same whatever the good, for instance a good of which you consume much or one which

has no interest for you. Xu would not accept the argument that you could become less free

because your favourite goods become more expensive, since this would depend on your

preferences (this is assumed to apply also to the case where the good you need most becomes

more expensive). However, this conception does not belong to all the actual uses of the terms

“free” and “freedom,” as we will see in conclusion.

Yet, the main problem may be with condition (1), Xu’s “invariance of scaling effects.”

This problem is that functions F and φ are not a priori unit-invariant. Indeed, the proposed

justification of condition (1) is that the ranking of freedom should not depend on the units in

which the quantities of goods are measured. If the unit of good i becomes λ times smaller,

then number ai should become λ times larger, and nothing real is changed. However, this is

12 More explicitly, condition (1) is the following. Let vector a take four values a1, a2, a3, a4, such that

3ia =λ 1

ia , 4ia =λ 2

ia , and, for all j≠i, 3ja = 1

ja and 4ja = 2

ja . Then φ(a1)≥φ(a2)⇒ φ (a3)≥ φ (a4) for any a1, a2, i, and λ>0 if and only if (2) holds.

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not described by condition (1) because, in fact, functions F and φ should change for taking

account of this change in unit. In technical terms, they incur the corresponding contravariant

transformation. Precisely because they represent real issues and not nominal ones. If one

wants to make explicit this issue of neutrality with respect to the choice of units of

measurement, one has to order not sets a={ai} of the numbers ai, but sets of numbers {ai/bi}

where bi is a given (arbitrary) quantity of the good i: when units change, both ai and bi are

multiplied by the same number, and ai/bi does not change (bi is a real – not nominal – unit of

measure of good i). An example will be met shortly.13

This can also be seen in another strong consequence of the proof under discussion,

namely, all utility functions are Cobb-Douglas provided it does not matter whether bread is

measured in kilos or in grams (and the like). Indeed, assume the ai denote now the quantities

of goods i consumed by an individual, and assume that the ordering under consideration is the

preference ordering of this individual. Consider Xu’s axioms except “symmetry.” The

individual prefers higher ai. Assume she is indifferent to the units of measure of the quantities

of all goods and similarly translate this as the “scaling effect.” Then, the individual’s ordinal

utility function has form (2), a specification of which is Π iiaα , a Cobb-Douglas utility

function.

3. Freedom as pointed distance 13 Xu works with orderings, and hence his orderings can incur the contravariant transformation. His ordering are representable by ordinal freedom functions. The kind of issue met is not unfrequent in normative economics. Another case has exactly the same structure. Suppose you want to justify Nash bargaining solution. Then, take φ to be a social welfare function, and ai=ui(x)–ui(x0) where ui is a cardinal utility function, x the social state, and x0 a particular reference state. Then ai is defined up to an arbitrary multiplicative factor independently for each i. If this is interpreted as implying condition (1), this leads to the form φ=φ1. A appeal to symmetry may then produce φ=φ2, and hence Πai as a specification of the social welfare function (Nash’s solution for n=2). Yet, this appeal to symmetry has no justification, and the appeal to condition (1) bypasses the contravariant transformation. In another story, φ is again a social welfare function, ai is a cardinal utility of individual i, and φ is also required to be cardinal (for instance, they are the corresponding von Neumann-Morgenstern specifications). Then, each of these functions being defined up to an increasing affine function, plus an appeal to symmetry, would require that φ is a utilitarian sum. Yet, this omits the contravariant transformation, and the symmetry has no justification (see Kolm 1996a, Chap. 14, and Maskin’s derivation of utilitarianism).A similar fallacy underlies the argument, which has been proposed, that an index of income inequality should be “scale-invariant” or homogeneous of degree zero because it should not change if the unit of measure of incomes changes (measure in dollars or in cents). Yet, an other index need not be unit-neutral and can incur the contravariant transformation when the unit for measuring incomes change. Note that the very term “scale-invariant” may suggest the mistake (homogeneity of degree zero as a real property, that is dependence on ratios – here of incomes – only, characterizes the measures that engineers and physicists call “intensive”).

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With this volume ranking of budget freedom, this freedom is also ranked as Πai and hence as

(Πai)1/n=y/(Πpi)1/n. This expression is homogeneous of degree zero in y and the pi. The

denominator – the geometric mean of the prices – is homogeneous of degree one in the pi. It is

a kind of price index and, then, the expression is a measure of real income.

More generally, a price index is a linearly homogeneous function of prices π(p), and

y/π(p) is the corresponding real income. Budget freedom is ranked by real income when the

freedom function F can be written as

F(y,p)=ϕ[y, π(p)],

since the homogeneity of degree zero of F in y and the pi implies

ϕ[y, π(p)]=ϕ[y/π(p),1]=f[y/π(p)]

where f is an increasing function. Since real income is also called purchasing power, budget

freedom is ranked as the corresponding purchasing power.

The volume ranking of budget freedoms takes the geometric mean of the prices as

price index (Kolm 2004a).

In contrast, the standard price index is a linear form π(p)=Σbipi with bi≥0 for all i and

bi>0 for at least some i. It seems that this is the only meaningful form of a price index,

because this is the cost of buying the set of quantities bi of the goods, the income necessary

for this purchase at these prices. It also seems that, for this reason, all the price indexes

actually used are applications of this form.

With this index, budget freedom is ranked as

β=y/Σbipi.

Hence,

Σβbipi=y.

This shows that the point of coordinates βbi is on the budget hyperplane of equation Σpixi=y.

If b={bi}denotes the vector of the coefficients of the price index bi, this point is βb.

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Hence, budget freedom is ranked as the distance to zero of the intersect of the budget

hyperplane with the ray from the origin bearing the vector b of the coefficients of the price

index (figure 1).

FIGURE 1

In particular, budget sets with equal budget freedom are budget sets whose budget

hyperplanes pass through the same point βb (figure 2).

FIGURE 2

Conversely, if a number of budget hyperplanes pass through the same point of the

non-negative orthant, they can be said to determine equal budget freedoms, relatively to a

vector of coefficients of the price index in the direction of this point.

That is, equal budget freedom corresponds to the pencils of budget hyperplanes.14

This result also amounts to the classical principle of free choice from an equal

(identical) allocation – which is represented by the common point of the budget

hyperplanes.15

However, the issue and result are important when situations with different prices are

compared. This can be different times, or places, or individuals when some prices are specific

to them (such as their wage rate which is the market price of their leisure or labour). The

choice of price indexes for comparing real income across times or places is the classical topic

of the theory of price indexes and of a vast literature. The corresponding question for

individuals and wage rates will shortly be noted.

14 With the general freedom function F(y, p), equal-freedom hyperplanes (the hyperplanes limiting equal-freedom budget sets) have an envelop, and these envelops for various freedoms do not intersect and are the graphs of quasi-concave functions which can be written as E(x)=F (see Appendix B). These envelops degenerate into points βb for a linear price index. They are rectangular hyperbolas for volume measures and n=2. 15 See Kolm 1971.

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The obtained form can illustrate the preceding remark about unit neutrality and

contravariant transformations. Indeed,

y/Σbi pi= (Σbi/ai)–1. (3)

Each coefficient bi has the dimension of a quantity of good i. When the units of good i are

divided by λ, both ai and bi are multiplied by λ, and expression (3) does not change.

The ranking of budget sets by their coefficients β permits the comparison of economic

situations with one budget set for each individual, say βj for individual j. The logic is that of

the comparison of profiles of co-ordinal indices (ordinal and interpersonnaly comparable).

The following properties are meaningful.16 Various counting of βj that are larger or smaller

than, or equal to, others. Equality of the βj (hence of the freedom of budget sets) in a situation,

and across situations for the same or different individuals. Pareto-like dominance.17

Permutation of the βj among individuals (symmetry of the evaluation if the permutation does

not affect it because the βj are the only relevant characteristics of the agents). “Fundamental”

Pareto-like dominance (Pareto-like dominance after some permutation of the βj in any state).

Truncations (increase in the lowests or decrease in the highests). Leximin.

It may also be interesting to be more specific than ordinal, and to consider a

conception of freedom which is not only ranked by real income but also measured by it. Then,

you would be twice freer if you are twice richer and hence if, with the same prices, you can

buy twice more of each good. These freedoms can be added and national or global freedom

would be national or global income. Inequality in freedom then is measured as income

inequality, and unfreedom as poverty.

4. Application: macrojustice as equal economic freedom

On ethical grounds, equal freedom of choice should be an aim when freedom of choice is the

only direct value of the relevant conception of justice. This value is freedom of choice rather

the outcome of the choice when the choosing individuals are deemed accountable for their

choice given their freedom of choice provided by their domains of possibilities. This

accountability is usually justified by their responsibility. Moreover, the direct value of justice

takes no account of any aspect of intensity of preferences in a number of cases. One of these 16 Kolm 1971, Part 3, and 2004a, Chapter 23. 17 I.e., the same comparison as Pareto’s for ordinal utilities.

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cases is very important: the overall distributive justice (in “macrojustice”) implemented by the

main fiscal tools such as the income tax. Indeed, nobody thinks that someone should pay a

higher income tax than someone else because she is less able to enjoy the euros taken away or

more able to enjoy the euros left (this discards both utilitarian and egalitarian welfarist

conceptions, and all intermediate ones). In these conditions, freedom of choice is the direct

value of justice. This implies that it should ideally be equal among the individuals, from a

reason of rationality in the basic sense of providing a reason, since there is no relevant item

which could justify non-equal solutions.18

However, the result obtained above is non-trivial only when prices differ among the

compared cases. For individuals and perfect competition, this happens when the wage rates

are among the prices, and hence leisure and labour are among the goods. This is precisely the

case for the determination of the just disposable income, just noted. The issue then is equal

freedom of choice – notably of work and earnings –, and the result consists of the

corresponding optimum transfers.

Consider m individuals indexed by i. There are two goods, income which enables one

to buy consumption goods, and leisure or labour. Individual i has income ci, leisure λi, labour

li, with λi+li=1 by choice of units, and given productivity and wage rate wi. Her earned

income is wili. She can receive transfer ti, which is a tax of –ti if ti<0.

Individual i’s income is

ci=wili+ti.

Her total income, including the value of leisure λi at its market price wi, is

yi=ci+wiλi=wi+ti

and corresponds to the income of the previous sections. This is the equation of individual i’s

budget line in the space of leisure λi and income ci (figure 3).

FIGURE 3

18 This logical necessity of prima facie or ideal equality in the relevant items is fully explained notably in Kolm 1998, Foreword, Section 5 (see also Kolm 1996a, Chapter 2).

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From the above, equal budget freedom implies that all budget lines pass through the

same point. Denote as k and η the labour and the income corresponding to this point,

respectively (the leisure is 1–k). We have, for all i,

η=wi k+ti.

Hence,

mη =kΣwi+Σti.

If the system is financially closed, Σti=0. Then, η=k w where w =(1/m) Σwi is average wage

rate, and ti=k·( w –wi).

The ti constitute a redistribution which amounts to redistributing equally the product of

the same labour k of all individuals (with their different productivities). This is equal labour

income equalization (ELIE). It produces equal pay for equal work for the equalization labour

k. It also amounts to the grant of an equal universal basic income (of wk ) to everyone,

financed by an equal sacrifice of each in terms of labour (k) – since each individual i then

pays wi k. Equivalently, this distribution amounts to each individual transferring to each other

the proceeds of the same labour of hers (k/m) – this is general balanced labour reciprocity.

The final outcome also amounts to all individuals freely choosing their labour and earnings

from an equal allocation of both leisure 1–k and income wk . Moreover, yi=wi+ti= wk +(1–

k)wi shows that the operation is a concentration of the total incomes (a linear uniform

concentration towards the mean) from their values yi=wi for k=0; this is the structure that most

uncontroversially reduces inequality.19

Since human capacities provide an extremely large part of the economic value of the

natural resources (labour provides the largest part of national income and it also largely

created capital), this ELIE solution allocates the bulk of resources. It constitutes overall

distributive justice, a part of macrojustice, along with social freedom. The case k=0 is

classical full self-ownership. The equalization labour k is a coefficient of equalization,

redistribution, community of human resources, reciprocity, and a minimum income in so far

as individuals are not responsible for their low income ( wkci > if ki >l , and wkci = if

wi=0).20 The amounts presently redistributed are those that would correspond to an

equalization labour of one to two days per week. One has ci= w k+wi·(li–k) and hence, with

19 Cf. Kolm 1999b. 20 Approximately, wk is to average earnings as k is to average labour duration.

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li>k for a normal full labour, the result amounts to individuals receiving according to their

work (desert) for the equalization labour k, and according to their work and capacities (merit)

for the rest of their labour.21

The prices are 1 for income and w for leisure-labour (i.e., wi for individual i). Since the

common possible allocation is wk for income with k for labour (1!k for leisure), the price

index can be taken as ywkwk =−+ )1( (yi for individual i, her total income). Its choice

amounts to the choice of the equalization labour or coefficient k, an issue which has been

analyzed in depth.22

The particular case k=ti=0 and an absence of transfers is very important because it

represents an essential historical social ethics, or ideology, and it is particularly important

here because its most common justification rests on a particular assumption about the

borderline between the various concepts of economic freedom. This is classical economic

liberalism promoting free market and free enterprise and rejecting distributive public

transfers. A higher wage rate wi then permits the individual to have a higher income

(consumption goods) for any given labour, or higher leisure for a given income, and hence

freedoms of choice for different productivities are unequal from the inclusion of domains.

Yet, this ethic classically justifies itself by freedom of action and exchange which apply basic

rights and negative or social freedom, and would be equally full for all. However, there can be

such freedoms with k≠0 and lump-sum transfers ti≠0. A classical liberal line of defense would

then be that if someone has to pay a net tax (such as a ti<0) out of earned income, she is in

fact forced to work to pay this liability, and forced labour is an infringement of social

(negative, protective) freedom. Hence, this latter freedom would finally preclude all

distributive transfers. However, the direct effect of this tax is that it reduces the freedom of

choice of bundles of income (goods) and leisure (in particular in precluding li=0 for a lump-

sum tax). Yet, a higher productivity (wi) by itself raises this freedom of choice since it enables

21 Kolm (2004a) provides the answers to the questions concerning : intensity of labour and effort; education and training; multidimensional labour; non-linear production functions of labour; involuntary unemployment; information about wage rates and given productivities; determination of the equalization labour and coefficient k; comparison with the relevant policies, proposals, and philosophies; method of practical implementation by reform of the present fiscal distributive tools; etc. See also Kolm (1993, 1996a, 1996b, and 2004b), and Maniquet (1998) whose question is a priori not freedom. 22 Part IV of Kolm 2004a.

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one to have more income with the same labour (or more leisure for the same income). Then,

the tax may only provide some compensation of this inequality if it is higher for more

productive individuals and subsidizes less productive ones. This means that the advantage or

handicap of having a higher or lower productivity is more or less “socialized” and shared.

This can only redistribute the value of the right to use one’s capacities, since this right itself

has to remain to the holder of the capacity if social freedom is respected. Classical economic

liberalism amounts to attributing also all this value to the holder of the capacity. That is, what

it really is is full self-ownership. This entails social freedom, but the converse does not hold

with the most cogent conception. These remarks constitute an essential part of economic and

social ethics.

5. Non-linear frontier

There is no reason to limit oneself to cases of agents facing given unit prices. The budget sets

often have other forms describing non-constant returns to scale, effects of scarcity, price

rebates, and so on. We continue to consider the standard economic case of divisible quantities

and free disposal. Hence, if a domain of possible choice includes { } nixx +ℜ∈= where xi is a

quantity of good i, it also includes all { } nixx +ℜ∈= '' where ii xx ≤≤ '0 for all i. A possibility

set is the closed subset nB +ℜ⊂ delimited by a n-dimensional hypersurface nP +ℜ⊂ of

equation P(x)=0 with ΜP/Μxi≥0 for all i. That is, B={x: nx +ℜ∈ and [ Px ∈∃ ' : x� 'x ]}.

The foregoing result says that, when P is a hyperplane, the freedom of choice provided

by the possibility set B is ranked (and, possibly, measured) by the largest multiple of a given

bundle b you can choose. That is, by the number β such that βb∈P. Since the exclusive focus

is on the points of the ray from the origin αb for all α>0, a priori the same ranking (or

measure) can be adopted in this case when P is not a hyperplane. One then has

ββ ≤⇒⊆ '' BB , and hence )()'( BFBF ≤ (yet, it is possible that both BB ⊂' and ββ =' ,

since points αb only are relevant for F).

Equal freedom of several possibility sets again mean that the frontiers P have a

common point on the ray αb.

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In the case of overall distribution in macrojustice, where equal freedom of choice is

the relevant principle as noted above, the production function of individual i is the income

fi(Ρi) she can have with labour ΡI, which may not be of the form wiΡi. The possibility frontier

is iiii tfy += )(l where ti �0 is a distributive transfer, with Σti=0. Individual i freely chooses

her labour Ρi and income yi in this domain. Equal freedom means that the frontiers have a

common point with Ρi=k for all i, and therefore iii tkfy +== )(η for all i, whence

)(kfm iΣ=η and )()()/1( kfkfm i =Σ=η , where m is the number of individuals. Then, the

distributive transfers are )()( kfkft ii −= . This is again an equal labour income equalization

with the equalization labour k. This also is, again, the universal allocation of the same basic

income )(kf financed by an equal sacrifice of labour k by each (i.e., a production fi(k) for

individual i). The determination of k (that is, of vector b of previous paragraphs) has been

analyzed in depth.23

This possible non-linearity of the constraint permits the consideration of constraints

bearing on quantitites. This is very important for the present application since this is the case

of involuntary unemployment where a constraint of the form 0ii ll ≤ , where 00 ≥il is a given

constant, is the available emplyment for individual i. Involuntary unemployment for

individual i is in general full if 0il =0 and only partial if 00 >= ii ll . Full involuntary

unemployment cannot be taken into account by the volume ranking of freedom, because it

gives a zero volume and hence a freedom inferior to all other cases, even if disposable income

yi is high because of large transfers. Involuntary unemployment can be taken into account in

replacing an impossoibility to work more (or at all) by an impossibility to earn more in

working more (or to earn in working). That is, function )( iif l is “truncated” in being

replaced by )( 0iif l for all 0

ii ll > . The result is that )(kfyi = if individual i is involuntarily

unemployed and ki ≤0l (hence kii ≤= 0ll ), and in particular if she is fully involuntarily

unemployed ( 00 == ii ll ).

Finally,all the foregoing extends to the case where labour Ρi, and hence the

equalization labour k, are multidimensional (duration, education and training, intensity, etc.),

with the same formulation. 23 See Kolm 2004a, Part IV.

16

6. Conclusion

The considerations of this note belong to the issue of characterizing freedom independently of

the preferences of the actor, as Xu emphasizes.24 This is one topic in freedom studies, but not

the only one. In other cases, considering preferences is essential. This is for instance the case

for potential freedom where the allocations are compared by the inclusion comparison of

possible domains of choice that can lead to choosing them.25 More generally, freedom is

relative to a set of constraints and possibilities, and it may be relevant, or not, to include

personal characteristics in them, notably properties of capacities to enjoy or be satisfied

(eudemonistic capacities) determined by tastes and influencing preferences or represented by

them. This depends on what one wants to do with the concept – note that Sections 4 and 5 do

not consider such capacities but consider productive personal capacities (the wage rate). One

can even consider the “freedom to obtain a given utility level” where the freedom functions

(specific to each individual) are the indirect (Roy) utility functions for linear (constant-price)

budget constraints. A similar representation holds for a production function in the space of

obtainable inputs for the “freedom to obtain a given output.” More generally, there even is a

case for identifying freedom with happiness since increased freedom that enables you to

choose alternatives that you enjoy more can make you happier, whereas unhappiness is

undesired and hence is a constraint. This equivalence (in the form of an equivalence between

liberation and the diminishing of insatisfaction or suffering) is a basis of the philosophical

psychology of advanced buddhism.26 Relatedly, mental freedom implies more or less

choosing one’s principles of action, desires – and hence preferences –, emotions, etc. As a

general rule, the relevant specification of the concept of freedom depends on what one wants

to do with it. In particular, discarding preferences has to be justified for each problem under

consideration. We have noted that this is the case for an issue of major importance, since this

discarding is a unanimous opinion for overall distributive justice in macrojustice (for instance

24 The next general step consists of characterizing fredom of choice when the actions chosen by various individuals can interfere, that is, the domain of choice of one agent can depend on the actions chosen by others – a “game-theoretic” situation. Normative considerations notably demand to define equal freedom in this case. The basic concept turns out to be symmetrical reciprocity, that is: “if I can choose a when you choose b, then you can choose a when I choose b”, equivalent to the symmetry of the social possibility set in the space of the actions of all agents (and symmetrical game forms). (See Kolm 1993). 25 Kolm 1999a. 26 See Kolm 1982.

17

for the income tax): de facto, everybody thinks that direct reference to a kind of “utility” is

irrelevant for this topic. This unanimous view should probably be endorsed, and at any rate it

will be implemented since everybody shares this opinion. Yet, various aspects of preferences

are relevant for more specific issues, and this can be associated with aspects of freedom.

7. Appendix. Extension to related concepts of budget freedom

The foregoing concepts extend in various ways.

1) Potential budget freedom

Let ijx the quantity of good j held by individual i, { } n

jij

i xx +ℜ∈= individual i’s allocation,

and { }ixX = the total allocation. We want to judge X, and the following ethics is a possible

one which is sometimes relevant. The direct value of justice is budget freedom, yet the

individuals’ choices can be potential only. Let ui(xi) be an ordinal quasi-concave utility

function of individual i, non-decreasing in ijx for all j. For a given xi, an implicit budget

hyperplane Hi(xi) is an hyperplane tangent to the hypersurface ui(ξ i)= ui(xi) where the

{ } nij

i+ℜ∈= ξξ are individual i’s allocations, and an implicit budget set Bi(xi) is the set of ξ i

such that there exists a )(' ii xH∈ξ such that ij

ij ξξ ≥' for all j, and 0≥i

jξ for all j. Hence, xi

is one of the possible choices of individual i if she were choosing in the budget set Bi(xi).

The freedom provided by the domain of choice Bi(xi) is the implicit or potential budget

freedom of individual i endowed with allocation xi. It is sometimes relevant to judge the total

allocation X in taking potential freedom as the direct value of justice.

Then, with the foregoing concepts and a linear price index of coefficients bj whose

vector is { }jbb = , the intersection of hyperplane Hi(xi) with the ray from the origin in the

direction of b is bxii ⋅)(β . The numbers βi(xi) provide an ordinal ranking of potential budget

freedoms, both across individuals i and across states x. This defines potential budget freedoms

that are equal or larger, and permits comparing total allocations by relations of dominance,

fundamental dominance, truncations, leximin, etc. (see Kolm 1971, Part 3, and 2004a,

18

Chapter 23). If one accepts to measure freedom by real income, the βi are these measures, and

all the comparisons of income distributions or inequalities can be used. However, these

comparisons or measures of freedom depend on the chosen price index coefficients b, and one

can have ui( 'ix )> ui(xi) with βi( 'ix )<βi(xi) (yet, individual i is then accountable for her

preferences and hence for her utility function).

2) More general freedom functions and indexes, characteristic envelops

With an ordinal freedom function F(D), the domains D that provide equal freedom constitute

a family. With a D defined in the space of bundles of quantities of goods j, { }jxx = , they

have an envelop. With linear budget freedom (facing constant prices p) with freedom function

F(y,p), the points of this envelop turn out to be with xj=!Fj/Fy if F is differentiable with

Fy=ΜF/Μy and Fj=ΜF/Μpj. The envelops corresponding each to a level of freedom have no

common point, with more freedom towards higher quantities, and form the lower boundary of

convex sets (they are representable by a quasi-concave function ⌡(x;F)=0 for each level of F,

which can be written as a non-decreasing quasi-concave function E(x)=F(y,p), with

gradE=p/y).

With a general price index π(p), and hence budget freedoms ordered as the real income

η=y/π(p), the point on the envelop for prices p has coordinates xj=ηπj where πj=Μπ/Μpj ,

assumed to exist, and hence is x=η grad π. A resulting property is that, at this point, the share

of the value of the good j in income is pjxj/y=pjπj/π , that is, the elasticity of the price index for

price pj.

For a linear price index π=Σbi pi, πj=bj and the envelop reduces to the common point

ηb of the budget hyperplanes of equal-freedom budget sets. The share of good j in income at

this allocation then is the share of its price in the price index, iijjjj pbpbyxp Σ= // .

With a geometric mean as price index, that is, a volume ranking of budget freedom,

there comes nyxp jj /= where n is the number of goods: at each point of the envelop, the

value of all goods have an equal share of the budget (for n=2, the equal-freedom envelops are

rectangular hyperbolas).

19

Both the linear index and the volume ranking are particular cases of a price index of

the form [ ] ααπ/1

)( ii pbΣ= when, respectively, α=1 and α 6 4. Then,

)/()/()/( 111jjjjjjjj pbppbpb πππππ αααααα === −−− ,

and the share of the value of the good j in income on the envelop is αααπ )(/)()/(/ iijjjjjj bpbpbpyxp Σ== .

Another extension of the volume takes iipp γπ Π=)( with 0>iγ for all i and 1=Σ iγ ,

and budget freedoms are ranked as iiaγΠ . The share of good j in income on the envelop is

jjj yxp γ=/ . The volume case takes nj /1=γ .

However, as far as tangible (economic) meaning is concerned – in contrast with

mathematical meaning –, it seems that linear indices only can really make sense.

References

Berlin I (1958) Two Concepts of Freedom. Clarendon Press, Oxford.

Elster J (1985) ed. The Multiple Self, Cambridge University Press, Cambridge.

Kolm S-Ch (1971) Justice et équité, Cepremap ; (1972) CNRS, Paris.

Kolm S-Ch (1982) Le Bonheur-Liberté (Bouddhisme profond et modernité). Presses Universitaires de France, Paris. Second edition, 1994.

Kolm S-Ch (1985) The Buddhist theory of “no-self.” In J. Elster,The Multiple Self, pp. 233-263.

Kolm S-Ch (1993) Equal liberty, CGPC, Paris.

Kolm S-Ch (1996a) Modern theories of justice. MIT Press, Cambridge, MA.

Kolm S-Ch (1996b) The theory of justice. Social Choice and Welfare 13: 151-82.

Kolm S-Ch (1998b) Justice and Equity, MIT Press, Cambridge, MA.

Kolm S-Ch (1999a) Freedom justice, CREME, University of Caen, 99-5.

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Kolm S-Ch (1999b) Rational foundation of income inequality measurement. In J. Silber, Handbook on Income Inequality Measurement, Kluwer Academic Publishers, London, pp.19-99.

Kolm S-Ch (2004a) Macrojustice, The political economy of fairness. Cambridge University Press, Cambridge.

Kolm S-Ch (2004b) Liberty and distribution: Macrojustice from social freedom. Social Choice and Welfare 22(1): 113-46.

Maniquet F (1998) An equal right solution to the compensation-responsibility dilemma. Math Soc Sci. 35: 185-201.

Pattanaik P and Y Xu (2000) On ranking opportunity sets in economic environments. J Econ Theory 93(1): 48-71.

Xu Y (2004) On ranking linear budget sets in terms of freedom of choice. Social Choice and Welfare, 22(1), 281-289.

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